Feature engineering for electricity load forecasting

The purpose of this notebook is to demonstrate how to use skrub and polars to perform feature engineering for electricity load forecasting.

We will build a set of features (and targets) from different data sources:

• Historical weather data for 10 medium to large urban areas in France;

• Holidays and standard calendar features for France;

• Historical electricity load data for the whole of France.

All these data sources cover a time range from March 23, 2021 to May 31, 2025.

Since our maximum forecasting horizon is 24 hours, we consider that the future weather data is known at a chosen prediction time. Similarly, the holidays and calendar features are known at prediction time for any point in the future.

Therefore, exogenous features derived from the weather and calendar data can be used to engineer “future covariates”. Since the load data is our prediction target, we will can also use it to engineer “past covariates” such as lagged features and rolling aggregations. The future values of the load data (with respect to the prediction time) are used as targets for the forecasting model.

Environment setup

We need to install some extra dependencies for this notebook if needed (when running jupyterlite). We need the development version of skrub to be able to use the skrub expressions.

%pip install -q https://pypi.anaconda.org/ogrisel/simple/polars/1.24.0/polars-1.24.0-cp39-abi3-emscripten_3_1_58_wasm32.whl
%pip install -q https://pypi.anaconda.org/ogrisel/simple/skrub/0.6.dev0/skrub-0.6.dev0-py3-none-any.whl
%pip install -q altair holidays plotly nbformat

ERROR: polars-1.24.0-cp39-abi3-emscripten_3_1_58_wasm32.whl is not a supported wheel on this platform.

Note: you may need to restart the kernel to use updated packages.

Note: you may need to restart the kernel to use updated packages.

Note: you may need to restart the kernel to use updated packages.

The following 3 imports are only needed to workaround some limitations when using polars in a pyodide/jupyterlite notebook.

TODO: remove those workarounds once pyodide 0.28 is released with support for the latest polars version.

import datetime
import tzdata  # noqa: F401
import pandas as pd
from pyarrow.parquet import read_table

import altair
import numpy as np
import polars as pl
import skrub
from pathlib import Path
import holidays
import warnings

from plotly.io import write_json, read_json  # noqa: F401

from tutorial_helpers import (
    binned_coverage,
    plot_lorenz_curve,
    plot_reliability_diagram,
    plot_residuals_vs_predicted,
    plot_binned_residuals,
    plot_horizon_forecast,
    collect_cv_predictions,
)

# Ignore warnings from pkg_resources triggered by Python 3.13's multiprocessing.
warnings.filterwarnings("ignore", category=UserWarning, module="pkg_resources")

Shared time range for all historical data sources

Let’s define a hourly time range from March 23, 2021 to May 31, 2025 that will be used to join the electricity load data and the weather data. The time range is in UTC timezone to avoid any ambiguity when joining with the weather data that is also in UTC.

We wrap the resulting polars dataframe in a skrub expression to benefit from the built-in skrub.TableReport display in the notebook. Using the skrub expression system will also be useful for other reasons: all operations in this notebook chain operations chained together in a directed acyclic graph that is automatically tracked by skrub. This allows us to extract the resulting pipeline and apply it to new data later on, exactly like a trained scikit-learn pipeline. The main difference is that we do so incrementally and while eagerly executing and inspecting the results of each step as is customary when working with dataframe libraries such as polars and pandas in Jupyter notebooks.

historical_data_start_time = skrub.var(
    "historical_data_start_time", pl.datetime(2021, 3, 23, hour=0, time_zone="UTC")
)
historical_data_end_time = skrub.var(
    "historical_data_end_time", pl.datetime(2025, 5, 31, hour=23, time_zone="UTC")
)

@skrub.deferred
def build_historical_time_range(
    historical_data_start_time,
    historical_data_end_time,
    time_interval="1h",
    time_zone="UTC",
):
    """Define an historical time range shared by all data sources."""
    return pl.DataFrame().with_columns(
        pl.datetime_range(
            start=historical_data_start_time,
            end=historical_data_end_time,
            time_zone=time_zone,
            interval=time_interval,
        ).alias("time"),
    )


time = build_historical_time_range(historical_data_start_time, historical_data_end_time)
time

<Call 'build_historical_time_range'>

Show graph

Result:

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If you run the above locally with pydot and graphviz installed, you can visualize the expression graph of the time variable by expanding the “Show graph” button.

Let’s now load the data records for the time range defined above.

To avoid network issues when running this notebook, the necessary data files have already been downloaded and saved in the datasets folder. See the README.md file for instructions to download the data manually if you want to re-run this notebook with more recent data.

data_source_folder = skrub.var("data_source_folder", Path("../datasets"))

for data_file in sorted(data_source_folder.skb.eval().iterdir()):
    print(data_file)

../datasets/README.md
../datasets/Total Load - Day Ahead _ Actual_202101010000-202201010000.csv
../datasets/Total Load - Day Ahead _ Actual_202201010000-202301010000.csv
../datasets/Total Load - Day Ahead _ Actual_202301010000-202401010000.csv
../datasets/Total Load - Day Ahead _ Actual_202401010000-202501010000.csv
../datasets/Total Load - Day Ahead _ Actual_202501010000-202601010000.csv
../datasets/weather_bayonne.parquet
../datasets/weather_brest.parquet
../datasets/weather_lille.parquet
../datasets/weather_limoges.parquet
../datasets/weather_lyon.parquet
../datasets/weather_marseille.parquet
../datasets/weather_nantes.parquet
../datasets/weather_paris.parquet
../datasets/weather_strasbourg.parquet
../datasets/weather_toulouse.parquet

We define a list of 10 medium to large urban areas to approximately cover most regions in France with a slight focus on most populated regions that are likely to drive electricity demand.

city_names = skrub.var(
    "city_names",
    [
        "paris",
        "lyon",
        "marseille",
        "toulouse",
        "lille",
        "limoges",
        "nantes",
        "strasbourg",
        "brest",
        "bayonne",
    ],
)


@skrub.deferred
def load_weather_data(time, city_names, data_source_folder):
    """Load and horizontal stack historical weather forecast data for each city."""
    all_city_weather = time
    for city_name in city_names:
        all_city_weather = all_city_weather.join(
            pl.from_arrow(
                read_table(f"{data_source_folder}/weather_{city_name}.parquet")
            )
            .with_columns([pl.col("time").dt.cast_time_unit("us")])
            .rename(lambda x: x if x == "time" else "weather_" + x + "_" + city_name),
            on="time",
        )
    return all_city_weather


all_city_weather = load_weather_data(time, city_names, data_source_folder)
all_city_weather

<Call 'load_weather_data'>

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Result:

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Calendar and holidays features

We leverage the holidays package to enrich the time range with some calendar features such as public holidays in France. We also add some features that are useful for time series forecasting such as the day of the week, the day of the year, and the hour of the day.

Note that the holidays package requires us to extract the date for the French timezone.

Similarly for the calendar features: all the time features are extracted from the time in the French timezone, since it is likely that electricity usage patterns are influenced by inhabitants’ daily routines aligned with the local timezone.

@skrub.deferred
def prepare_french_calendar_data(time):
    fr_time = pl.col("time").dt.convert_time_zone("Europe/Paris")
    fr_year_min = time.select(fr_time.dt.year().min()).item()
    fr_year_max = time.select(fr_time.dt.year().max()).item()
    holidays_fr = holidays.country_holidays(
        "FR", years=range(fr_year_min, fr_year_max + 1)
    )
    return time.with_columns(
        [
            fr_time.dt.hour().alias("cal_hour_of_day"),
            fr_time.dt.weekday().alias("cal_day_of_week"),
            fr_time.dt.ordinal_day().alias("cal_day_of_year"),
            fr_time.dt.year().alias("cal_year"),
            fr_time.dt.date().is_in(holidays_fr.keys()).alias("cal_is_holiday"),
        ],
    )


calendar = prepare_french_calendar_data(time)
calendar

<Call 'prepare_french_calendar_data'>

Show graph

Result:

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Electricity load data

Finally we load the electricity load data. This data will both be used as a target variable but also to craft some lagged and window-aggregated features.

@skrub.deferred
def load_electricity_load_data(time, data_source_folder):
    """Load and aggregate historical load data from the raw CSV files."""
    load_data_files = [
        data_file
        for data_file in sorted(data_source_folder.iterdir())
        if data_file.name.startswith("Total Load - Day Ahead")
        and data_file.name.endswith(".csv")
    ]
    return time.join(
        (
            pl.concat(
                [
                    pl.from_pandas(pd.read_csv(data_file, na_values=["N/A", "-"])).drop(
                        ["Day-ahead Total Load Forecast [MW] - BZN|FR"]
                    )
                    for data_file in load_data_files
                ]
            ).select(
                [
                    pl.col("Time (UTC)")
                    .str.split(by=" - ")
                    .list.first()
                    .str.to_datetime("%d.%m.%Y %H:%M", time_zone="UTC")
                    .alias("time"),
                    pl.col("Actual Total Load [MW] - BZN|FR").alias("load_mw"),
                ]
            )
        ),
        on="time",
    )

Let’s load the data and check if there are missing values since we will use this data as the target variable for our forecasting model.

electricity_raw = load_electricity_load_data(time, data_source_folder)
electricity_raw.filter(pl.col("load_mw").is_null())

<CallMethod 'filter'>

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Result:

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So apparently there a few missing measurements. Let’s use linear interpolation to fill those missing values.

electricity_raw.filter(
    (pl.col("time") > pl.datetime(2021, 10, 30, hour=10, time_zone="UTC"))
    & (pl.col("time") < pl.datetime(2021, 10, 31, hour=10, time_zone="UTC"))
).skb.eval().plot.line(x="time:T", y="load_mw:Q")

electricity = electricity_raw.with_columns([pl.col("load_mw").interpolate()])
electricity.filter(
    (pl.col("time") > pl.datetime(2021, 10, 30, hour=10, time_zone="UTC"))
    & (pl.col("time") < pl.datetime(2021, 10, 31, hour=10, time_zone="UTC"))
).skb.eval().plot.line(x="time:T", y="load_mw:Q")

Remark: interpolating missing values in the target column that we will use to train and evaluate our models can bias the learning problem and make our cross-validation metrics misrepresent the performance of the deployed predictive system.

A potentially better approach would be to keep the missing values in the dataset and use a sample_weight mask to keep a contiguous dataset while ignoring the time periods with missing values when training or evaluating the model.

Lagged features

We can now create some lagged features from the electricity load data.

We will create 3 hourly lagged features, 1 daily lagged feature, and 1 weekly lagged feature. We will also create a rolling median and inter-quartile feature over the last 24 hours and over the last 7 days.

def iqr(col, *, window_size: int):
    """Inter-quartile range (IQR) of a column."""
    return col.rolling_quantile(0.75, window_size=window_size) - col.rolling_quantile(
        0.25, window_size=window_size
    )


electricity_lagged = electricity.with_columns(
    [pl.col("load_mw").shift(i).alias(f"load_mw_lag_{i}h") for i in range(1, 4)]
    + [
        pl.col("load_mw").shift(24).alias("load_mw_lag_1d"),
        pl.col("load_mw").shift(24 * 7).alias("load_mw_lag_1w"),
        pl.col("load_mw")
        .rolling_median(window_size=24)
        .alias("load_mw_rolling_median_24h"),
        pl.col("load_mw")
        .rolling_median(window_size=24 * 7)
        .alias("load_mw_rolling_median_7d"),
        iqr(pl.col("load_mw"), window_size=24).alias("load_mw_iqr_24h"),
        iqr(pl.col("load_mw"), window_size=24 * 7).alias("load_mw_iqr_7d"),
    ],
)
electricity_lagged

<CallMethod 'with_columns'>

Show graph

Result:

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altair.Chart(electricity_lagged.tail(100).skb.preview()).transform_fold(
    [
        "load_mw",
        "load_mw_lag_1h",
        "load_mw_lag_2h",
        "load_mw_lag_3h",
        "load_mw_lag_1d",
        "load_mw_lag_1w",
        "load_mw_rolling_median_24h",
        "load_mw_rolling_median_7d",
        "load_mw_rolling_iqr_24h",
        "load_mw_rolling_iqr_7d",
    ],
    as_=["key", "load_mw"],
).mark_line(tooltip=True).encode(x="time:T", y="load_mw:Q", color="key:N").interactive()

Important remark about lagged features engineering and system lag

When working with historical data, we often have access to all the past measurements in the dataset. However, when we want to use the lagged features in a forecasting model, we need to be careful about the length of the system lag: the time between a timestamped measurement is made in the real world and the time the record is made available to the downstream application (in our case, a deployed predictive pipeline).

System lag is rarely explicitly represented in the data sources even if such delay can be as large as several hours or even days and can sometimes be irregular. For instance, if there is a human intervention in the data recording process, holidays and weekends can punctually add significant delay.

If the system lag is larger than the maximum feature engineering lag, the resulting features be filled with missing values once deployed. More importantly, if the system lag is not handled explicitly, those resulting missing values will only be present in the features computed for the deployed system but not present in the features computed to train and backtest the system before deployment.

This structural discrepancy can severely degrade the performance of the deployed model compared to the performance estimated from backtesting on the historical data.

We will set this problem aside for now but discuss it again in a later section of this tutorial.

Investigating outliers in the lagged features

Let’s use the skrub.TableReport tool to look at the plots of the marginal distribution of the lagged features.

from skrub import TableReport

TableReport(electricity_lagged.skb.eval())

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Let’s extract the dates where the inter-quartile range of the load is greater than 15,000 MW.

electricity_lagged.filter(pl.col("load_mw_iqr_7d") > 15_000)[
    "time"
].dt.date().unique().sort().to_list().skb.eval()

[datetime.date(2021, 12, 26),
 datetime.date(2021, 12, 27),
 datetime.date(2021, 12, 28),
 datetime.date(2022, 1, 7),
 datetime.date(2022, 1, 8),
 datetime.date(2023, 1, 19),
 datetime.date(2023, 1, 20),
 datetime.date(2023, 1, 21),
 datetime.date(2024, 1, 10),
 datetime.date(2024, 1, 11),
 datetime.date(2024, 1, 12),
 datetime.date(2024, 1, 13)]

We observe 3 date ranges with high inter-quartile range. Let’s plot the electricity load and the lagged features for the first data range along with the weather data for Paris.

altair.Chart(
    electricity_lagged.filter(
        (pl.col("time") > pl.datetime(2021, 12, 1, time_zone="UTC"))
        & (pl.col("time") < pl.datetime(2021, 12, 31, time_zone="UTC"))
    ).skb.eval()
).transform_fold(
    [
        "load_mw",
        "load_mw_iqr_7d",
    ],
).mark_line(
    tooltip=True
).encode(
    x="time:T", y="value:Q", color="key:N"
).interactive()

altair.Chart(
    all_city_weather.filter(
        (pl.col("time") > pl.datetime(2021, 12, 1, time_zone="UTC"))
        & (pl.col("time") < pl.datetime(2021, 12, 31, time_zone="UTC"))
    ).skb.eval()
).transform_fold(
    [f"weather_temperature_2m_{city_name}" for city_name in city_names.skb.eval()],
).mark_line(
    tooltip=True
).encode(
    x="time:T", y="value:Q", color="key:N"
).interactive()

Based on the plots above, we can see that the electricity load was high just before the Christmas holidays due to low temperatures. Then the load suddenly dropped because temperatures went higher right at the start of the end-of-year holidays.

So those outliers do not seem to be caused to a data quality issue but rather due to a real change in the electricity load demand. We could conduct similar analysis for the other date ranges with high inter-quartile range but we will skip that for now.

If we had observed significant data quality issues over extended periods of time could have been addressed by removing the corresponding rows from the dataset. However, this would make the lagged and windowing feature engineering challenging to reimplement correctly. A better approach would be to keep a contiguous dataset assign 0 weights to the affected rows when fitting or evaluating the trained models via the use of the sample_weight parameter.

Final dataset

We now assemble the dataset that will be used to train and evaluate the forecasting models via backtesting.

prediction_start_time = skrub.var(
    "prediction_start_time", historical_data_start_time.skb.eval() + pl.duration(days=7)
)
prediction_end_time = skrub.var(
    "prediction_end_time", historical_data_end_time.skb.eval() - pl.duration(hours=24)
)


@skrub.deferred
def define_prediction_time_range(prediction_start_time, prediction_end_time):
    return pl.DataFrame().with_columns(
        pl.datetime_range(
            start=prediction_start_time,
            end=prediction_end_time,
            time_zone="UTC",
            interval="1h",
        ).alias("prediction_time"),
    )


prediction_time = define_prediction_time_range(
    prediction_start_time, prediction_end_time
).skb.subsample(n=1000, how="head")
prediction_time

<SubsamplePreviews>

Show graph

Result (on a subsample):

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@skrub.deferred
def build_features(
    prediction_time,
    electricity_lagged,
    all_city_weather,
    calendar,
    future_feature_horizons=[1, 24],
):

    return (
        prediction_time.join(
            electricity_lagged, left_on="prediction_time", right_on="time"
        )
        .join(
            all_city_weather.select(
                [pl.col("time")]
                + [
                    pl.col(c).shift(-h).alias(c + f"_future_{h}h")
                    for c in all_city_weather.columns
                    if c != "time"
                    for h in future_feature_horizons
                ]
            ),
            left_on="prediction_time",
            right_on="time",
        )
        .join(
            calendar.select(
                [pl.col("time")]
                + [
                    pl.col(c).shift(-h).alias(c + f"_future_{h}h")
                    for c in calendar.columns
                    if c != "time"
                    for h in future_feature_horizons
                ]
            ),
            left_on="prediction_time",
            right_on="time",
        )
    ).drop("prediction_time")


features = build_features(
    prediction_time=prediction_time,
    electricity_lagged=electricity_lagged,
    all_city_weather=all_city_weather,
    calendar=calendar,
).skb.mark_as_X()

features

<Call 'build_features'>

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Result (on a subsample):

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Let’s build training and evaluation targets for all possible horizons from 1 to 24 hours.

horizons = range(1, 25)
target_column_name_pattern = "load_mw_horizon_{horizon}h"


@skrub.deferred
def build_targets(prediction_time, electricity, horizons):
    return prediction_time.join(
        electricity.with_columns(
            [
                pl.col("load_mw")
                .shift(-h)
                .alias(target_column_name_pattern.format(horizon=h))
                for h in horizons
            ]
        ),
        left_on="prediction_time",
        right_on="time",
    )


targets = build_targets(prediction_time, electricity, horizons)
targets

<Call 'build_targets'>

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Result (on a subsample):

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For now, let’s focus on the last horizon (24 hours) to train a model predicting the electricity load at the next 24 hours.

horizon_of_interest = horizons[-1]  # Focus on the 24-hour horizon
target_column_name = target_column_name_pattern.format(horizon=horizon_of_interest)
predicted_target_column_name = "predicted_" + target_column_name
target = targets[target_column_name].skb.mark_as_y()
target

<GetItem 'load_mw_horizon_24h'>

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Result (on a subsample):

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Let’s define our first single output prediction pipeline. This pipeline chains our previous feature engineering steps with a skrub.DropCols step to drop some columns that we do not want to use as features, and a HistGradientBoostingRegressor model from scikit-learn.

The skrub.choose_from, skrub.choose_float, and skrub.choose_int functions are used to define hyperparameters that can be tuned via cross-validated randomized search.

from sklearn.ensemble import HistGradientBoostingRegressor
import skrub.selectors as s


features_with_dropped_cols = features.skb.apply(
    skrub.DropCols(
        cols=skrub.choose_from(
            {
                "none": s.glob(""),  # No column has an empty name.
                "load": s.glob("load_*"),
                "rolling_load": s.glob("load_mw_rolling_*"),
                "weather": s.glob("weather_*"),
                "temperature": s.glob("weather_temperature_*"),
                "moisture": s.glob("weather_moisture_*"),
                "cloud_cover": s.glob("weather_cloud_cover_*"),
                "calendar": s.glob("cal_*"),
                "holiday": s.glob("cal_is_holiday*"),
                "future_1h": s.glob("*_future_1h"),
                "future_24h": s.glob("*_future_24h"),
                "non_paris_weather": s.glob("weather_*") & ~s.glob("weather_*_paris_*"),
            },
            name="dropped_cols",
        )
    )
)

hgbr_predictions = features_with_dropped_cols.skb.apply(
    HistGradientBoostingRegressor(
        random_state=0,
        loss=skrub.choose_from(["squared_error", "poisson", "gamma"], name="loss"),
        learning_rate=skrub.choose_float(
            0.01, 1, default=0.1, log=True, name="learning_rate"
        ),
        max_leaf_nodes=skrub.choose_int(
            3, 300, default=30, log=True, name="max_leaf_nodes"
        ),
    ),
    y=target,
)
hgbr_predictions

<Apply HistGradientBoostingRegressor>

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Result (on a subsample):

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The predictions expression captures the whole expression graph that includes the feature engineering steps, the target variable, and the model training step.

In particular, the input data keys for the full pipeline can be inspected as follows:

hgbr_predictions.skb.get_data().keys()

dict_keys(['prediction_start_time', 'prediction_end_time', 'historical_data_start_time', 'historical_data_end_time', 'data_source_folder', 'city_names'])

Furthermore, the hyper-parameters of the full pipeline can be retrieved as follows:

hgbr_pipeline = hgbr_predictions.skb.get_pipeline()
hgbr_pipeline.describe_params()

{'dropped_cols': 'none',
 'learning_rate': 0.1,
 'loss': 'squared_error',
 'max_leaf_nodes': 30}

When running this notebook locally, you can also interactively inspect all the steps of the DAG using the following (once uncommented):

# predictions.skb.full_report()

Since we passed input values to all the upstream skrub variables, skrub automatically evaluates the whole expression graph graph (train and predict on the same data) so that we can interactively check that everything will work as expected.

Let’s use altair to visualize the predictions against the target values for the last 24 hours of the prediction time range used to train the model. This allows us can (over)fit the data with the features at hand.

altair.Chart(
    pl.concat(
        [
            targets.skb.preview(),
            hgbr_predictions.rename(
                {target_column_name: predicted_target_column_name}
            ).skb.preview(),
        ],
        how="horizontal",
    ).tail(24 * 7)
).transform_fold(
    [target_column_name, predicted_target_column_name],
).mark_line(
    tooltip=True
).encode(
    x="prediction_time:T", y="value:Q", color="key:N"
).interactive()

Assessing the model performance via cross-validation

Being able to fit the training data is not enough. We need to assess the ability of the training pipeline to learn a predictive model that can generalize to unseen data.

Furthermore, we want to assess the uncertainty of this estimate of the generalization performance via time-based cross-validation, also known as backtesting.

scikit-learn provides a TimeSeriesSplit splitter providing a convenient way to split temporal data: in the different folds, the training data always precedes the test data. It implies that the size of the training data is getting larger as the fold index increases. The scikit-learn utility allows to define a couple of parameters to control the size of the training and test data and as well as a gap between the training and test data to potentially avoid leakage if our model relies on lagged features.

In the example below, we define that the training data should be at most 2 years worth of data and the test data should be 24 weeks long. We also define a gap of 1 week between the training.

Let’s check those statistics by iterating over the different folds provided by the splitter.

from sklearn.model_selection import TimeSeriesSplit


max_train_size = 2 * 52 * 24 * 7  # max ~2 years of training data
test_size = 24 * 7 * 24  # 24 weeks of test data
gap = 7 * 24  # 1 week gap between train and test sets
ts_cv_5 = TimeSeriesSplit(
    n_splits=5, max_train_size=max_train_size, test_size=test_size, gap=gap
)

for fold_idx, (train_idx, test_idx) in enumerate(
    ts_cv_5.split(prediction_time.skb.eval())
):
    print(f"CV iteration #{fold_idx}")
    train_datetimes = prediction_time.skb.eval()[train_idx]
    test_datetimes = prediction_time.skb.eval()[test_idx]
    print(
        f"Train: {train_datetimes.shape[0]} rows, "
        f"Test: {test_datetimes.shape[0]} rows"
    )
    print(f"Train time range: {train_datetimes[0, 0]} to " f"{train_datetimes[-1, 0]} ")
    print(f"Test time range: {test_datetimes[0, 0]} to " f"{test_datetimes[-1, 0]} ")
    print()

CV iteration #0
Train: 16224 rows, Test: 4032 rows
Train time range: 2021-03-30 00:00:00+00:00 to 2023-02-03 23:00:00+00:00 
Test time range: 2023-02-11 00:00:00+00:00 to 2023-07-28 23:00:00+00:00 

CV iteration #1
Train: 17472 rows, Test: 4032 rows
Train time range: 2021-07-24 00:00:00+00:00 to 2023-07-21 23:00:00+00:00 
Test time range: 2023-07-29 00:00:00+00:00 to 2024-01-12 23:00:00+00:00 

CV iteration #2
Train: 17472 rows, Test: 4032 rows
Train time range: 2022-01-08 00:00:00+00:00 to 2024-01-05 23:00:00+00:00 
Test time range: 2024-01-13 00:00:00+00:00 to 2024-06-28 23:00:00+00:00 

CV iteration #3
Train: 17472 rows, Test: 4032 rows
Train time range: 2022-06-25 00:00:00+00:00 to 2024-06-21 23:00:00+00:00 
Test time range: 2024-06-29 00:00:00+00:00 to 2024-12-13 23:00:00+00:00 

CV iteration #4
Train: 17472 rows, Test: 4032 rows
Train time range: 2022-12-10 00:00:00+00:00 to 2024-12-06 23:00:00+00:00 
Test time range: 2024-12-14 00:00:00+00:00 to 2025-05-30 23:00:00+00:00 

Once the cross-validation strategy is defined, we pass it to the cross_validate function provided by skrub to compute the cross-validated scores. Here, we define the mean absolute percentage error that is interpretable. However, this metric is not a proper scoring rule. We therefore look at the R2 score and the Tweedie deviance score.

from sklearn.metrics import make_scorer, mean_absolute_percentage_error, get_scorer
from sklearn.metrics import d2_tweedie_score


hgbr_cv_results = hgbr_predictions.skb.cross_validate(
    cv=ts_cv_5,
    scoring={
        "mape": make_scorer(mean_absolute_percentage_error),
        "r2": get_scorer("r2"),
        "d2_poisson": make_scorer(d2_tweedie_score, power=1.0),
        "d2_gamma": make_scorer(d2_tweedie_score, power=2.0),
    },
    return_train_score=True,
    return_pipeline=True,
    verbose=1,
    n_jobs=-1,
)
hgbr_cv_results.round(3)

[Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.

[Parallel(n_jobs=-1)]: Done   5 out of   5 | elapsed:    8.6s finished

	fit_time	score_time	test_mape	train_mape	test_r2	train_r2	test_d2_poisson	train_d2_poisson	test_d2_gamma	train_d2_gamma	pipeline
0	2.966	0.062	0.027	0.012	0.963	0.994	0.962	0.994	0.961	0.994	SkrubPipeline(expr=<Apply HistGradientBoosting...
1	3.275	0.063	0.024	0.013	0.978	0.994	0.977	0.994	0.976	0.993	SkrubPipeline(expr=<Apply HistGradientBoosting...
2	3.232	0.064	0.023	0.014	0.974	0.993	0.974	0.993	0.975	0.992	SkrubPipeline(expr=<Apply HistGradientBoosting...
3	3.238	0.058	0.019	0.014	0.980	0.993	0.980	0.992	0.980	0.992	SkrubPipeline(expr=<Apply HistGradientBoosting...
4	2.157	0.038	0.023	0.014	0.977	0.993	0.978	0.992	0.978	0.992	SkrubPipeline(expr=<Apply HistGradientBoosting...

TODO: comment the results obtained via cross-validation.

We further analyze our cross-validated model by collecting the predictions on each split.

hgbr_cv_predictions = collect_cv_predictions(
    hgbr_cv_results["pipeline"], ts_cv_5, hgbr_predictions, prediction_time
)
hgbr_cv_predictions[0]

shape: (4_032, 3)

prediction_time	load_mw	predicted_load_mw
datetime[μs, UTC]	f64	f64
2023-02-11 00:00:00 UTC	59258.0	59855.334418
2023-02-11 01:00:00 UTC	58654.0	59958.654564
2023-02-11 02:00:00 UTC	56155.0	57666.184522
2023-02-11 03:00:00 UTC	54463.0	55832.880673
2023-02-11 04:00:00 UTC	54698.0	57121.984097
…	…	…
2023-07-28 19:00:00 UTC	38781.0	40093.987086
2023-07-28 20:00:00 UTC	38455.0	39343.771368
2023-07-28 21:00:00 UTC	39972.0	40738.151594
2023-07-28 22:00:00 UTC	39825.0	39449.468131
2023-07-28 23:00:00 UTC	36822.0	35828.293662

The first curve is called the Lorenz curve. It shows on the x-axis the fraction of observations sorted by predicted values and on the y-axis the cumulative observed load proportion.

plot_lorenz_curve(hgbr_cv_predictions).interactive()

The diagonal on the plot corresponds to a model predicting a constant value that is therefore not an informative model. The oracle model corresponds to the “perfect” model that would provide the an output identical to the observed values. Thus, the ranking of such hypothetical model is the best possible ranking. However, you should note that the oracle model is not the line passing through the right-hand corner of the plot. Instead, this curvature is defined by the distribution of the observations. Indeed, more the observations are composed of small values and a couple of large values, the more the oracle model is closer to the right-hand corner of the plot.

A true model is navigating between the diagonal and the oracle model. The area between the diagonal and the Lorenz curve of a model is called the Gini index.

For our model, we observe that each oracle model is not far from the diagonal. It means that the observed values do not contain a couple of large values with high variability. Therefore, it informs us that the complexity of our problem at hand is not too high. Looking at the Lorenz curve of each model, we observe that it is quite close to the oracle model. Therefore, the gradient boosting regressor is discriminative for our task.

Then, we have a look at the reliability diagram. This diagram shows on the x-axis the mean predicted load and on the y-axis the mean observed load.

plot_reliability_diagram(hgbr_cv_predictions).interactive().properties(
    title="Reliability diagram from cross-validation predictions"
)

The diagonal on the reliability diagram corresponds to the best possible model: for a level of predicted load that fall into a bin, then the mean observed load is also in the same bin. If the line is above the diagonal, it means that our model is predicted a value too low in comparison to the observed values. If the line is below the diagonal, it means that our model is predicted a value too high in comparison to the observed values.

For our cross-validated model, we observe that each reliability curve is close to the diagonal. We only observe a mis-calibration for the extremum values.

plot_residuals_vs_predicted(hgbr_cv_predictions).interactive().properties(
    title="Residuals vs Predicted Values from cross-validation predictions"
)

plot_binned_residuals(hgbr_cv_predictions, by="hour").interactive().properties(
    title="Residuals by hour of the day from cross-validation predictions"
)

plot_binned_residuals(hgbr_cv_predictions, by="month").interactive().properties(
    title="Residuals by hour of the day from cross-validation predictions"
)

ts_cv_2 = TimeSeriesSplit(
    n_splits=2, test_size=test_size, max_train_size=max_train_size, gap=24
)
# randomized_search_hgbr = hgbr_predictions.skb.get_randomized_search(
#     cv=ts_cv_2,
#     scoring="r2",
#     n_iter=100,
#     fitted=True,
#     verbose=1,
#     n_jobs=-1,
# )
# # %%
# randomized_search_hgbr.results_.round(3)

# fig = randomized_search_hgbr.plot_results().update_layout(margin=dict(l=200))
# write_json(fig, "parallel_coordinates_hgbr.json")

fig = read_json("parallel_coordinates_hgbr.json")
fig.update_layout(margin=dict(l=200))

# nested_cv_results = skrub.cross_validate(
#     environment=predictions.skb.get_data(),
#     pipeline=randomized_search,
#     cv=ts_cv_5,
#     scoring={
#         "r2": get_scorer("r2"),
#         "mape": make_scorer(mean_absolute_percentage_error),
#     },
#     n_jobs=-1,
#     return_pipeline=True,
# ).round(3)
# nested_cv_results

# for outer_fold_idx in range(len(nested_cv_results)):
#     print(
#         nested_cv_results.loc[outer_fold_idx, "pipeline"]
#         .results_.loc[0]
#         .round(3)
#         .to_dict()
#     )

Exercise: non-linear feature engineering coupled with linear predictive model

Now, it is your turn to make a predictive model. Towards this end, we request you to preprocess the input features with non-linear feature engineering:

• the first step is to impute the missing values using a SimpleImputer. Make sure to include the indicator of missing values in the feature set (i.e. look at the add_indicator parameter);

• use a SplineTransformer to create non-linear features. Use the default parameters but make sure to set sparse_output=True since it subsequent processing will be faster and more memory efficient with such data structure;

• use a VarianceThreshold to remove features with potential constant features;

• use a SelectKBest to select the most informative features. Set k to be chosen from a log-uniform distribution between 100 and 1,000 (i.e. use skrub.choose_int);

• use a Nystroem to approximate an RBF kernel. Set n_components to be chosen from a log-uniform distribution between 10 and 200 (i.e. use skrub.choose_int).

• finally, use a Ridge as the final predictive model. Set alpha to be chosen from a log-uniform distribution between 1e-6 and 1e3 (i.e. use skrub.choose_float).

Use a scikit-learn Pipeline using make_pipeline to chain the steps together.

Once the predictive model is defined, apply it on the feature_with_dropped_cols expression. Do not forget to define that target is the y variable.

# Here we provide all the imports for creating the predictive model.
from sklearn.feature_selection import SelectKBest, VarianceThreshold
from sklearn.impute import SimpleImputer
from sklearn.linear_model import Ridge
from sklearn.kernel_approximation import Nystroem
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import SplineTransformer

# Write your code here.
#
#
#
#
#
#
#
#
#
#
#

predictions_ridge = features_with_dropped_cols.skb.apply(
    make_pipeline(
        SimpleImputer(add_indicator=True),
        SplineTransformer(sparse_output=True),
        VarianceThreshold(threshold=1e-6),
        SelectKBest(
            k=skrub.choose_int(100, 1_000, log=True, name="n_selected_splines")
        ),
        Nystroem(
            n_components=skrub.choose_int(
                10, 200, log=True, name="n_components", default=150
            )
        ),
        Ridge(
            alpha=skrub.choose_float(1e-6, 1e3, log=True, name="alpha", default=1e-2)
        ),
    ),
    y=target,
)
predictions_ridge

/home/runner/work/forecasting/forecasting/.pixi/envs/doc/lib/python3.12/site-packages/sklearn/impute/_base.py:637: UserWarning:

Skipping features without any observed values: ['weather_soil_moisture_1_to_3cm_paris_future_1h'
 'weather_soil_moisture_1_to_3cm_paris_future_24h'
 'weather_soil_moisture_1_to_3cm_lyon_future_1h'
 'weather_soil_moisture_1_to_3cm_lyon_future_24h'
 'weather_soil_moisture_1_to_3cm_marseille_future_1h'
 'weather_soil_moisture_1_to_3cm_marseille_future_24h'
 'weather_soil_moisture_1_to_3cm_toulouse_future_1h'
 'weather_soil_moisture_1_to_3cm_toulouse_future_24h'
 'weather_soil_moisture_1_to_3cm_lille_future_1h'
 'weather_soil_moisture_1_to_3cm_lille_future_24h'
 'weather_soil_moisture_1_to_3cm_limoges_future_1h'
 'weather_soil_moisture_1_to_3cm_limoges_future_24h'
 'weather_soil_moisture_1_to_3cm_nantes_future_1h'
 'weather_soil_moisture_1_to_3cm_nantes_future_24h'
 'weather_soil_moisture_1_to_3cm_strasbourg_future_1h'
 'weather_soil_moisture_1_to_3cm_strasbourg_future_24h'
 'weather_soil_moisture_1_to_3cm_brest_future_1h'
 'weather_soil_moisture_1_to_3cm_brest_future_24h'
 'weather_soil_moisture_1_to_3cm_bayonne_future_1h'
 'weather_soil_moisture_1_to_3cm_bayonne_future_24h']. At least one non-missing value is needed for imputation with strategy='mean'.

/home/runner/work/forecasting/forecasting/.pixi/envs/doc/lib/python3.12/site-packages/sklearn/feature_selection/_univariate_selection.py:111: RuntimeWarning:

divide by zero encountered in divide

/home/runner/work/forecasting/forecasting/.pixi/envs/doc/lib/python3.12/site-packages/sklearn/impute/_base.py:637: UserWarning:

Skipping features without any observed values: ['weather_soil_moisture_1_to_3cm_paris_future_1h'
 'weather_soil_moisture_1_to_3cm_paris_future_24h'
 'weather_soil_moisture_1_to_3cm_lyon_future_1h'
 'weather_soil_moisture_1_to_3cm_lyon_future_24h'
 'weather_soil_moisture_1_to_3cm_marseille_future_1h'
 'weather_soil_moisture_1_to_3cm_marseille_future_24h'
 'weather_soil_moisture_1_to_3cm_toulouse_future_1h'
 'weather_soil_moisture_1_to_3cm_toulouse_future_24h'
 'weather_soil_moisture_1_to_3cm_lille_future_1h'
 'weather_soil_moisture_1_to_3cm_lille_future_24h'
 'weather_soil_moisture_1_to_3cm_limoges_future_1h'
 'weather_soil_moisture_1_to_3cm_limoges_future_24h'
 'weather_soil_moisture_1_to_3cm_nantes_future_1h'
 'weather_soil_moisture_1_to_3cm_nantes_future_24h'
 'weather_soil_moisture_1_to_3cm_strasbourg_future_1h'
 'weather_soil_moisture_1_to_3cm_strasbourg_future_24h'
 'weather_soil_moisture_1_to_3cm_brest_future_1h'
 'weather_soil_moisture_1_to_3cm_brest_future_24h'
 'weather_soil_moisture_1_to_3cm_bayonne_future_1h'
 'weather_soil_moisture_1_to_3cm_bayonne_future_24h']. At least one non-missing value is needed for imputation with strategy='mean'.

<Apply Pipeline>

Show graph

Result (on a subsample):

Please enable javascript

The skrub table reports need javascript to display correctly. If you are displaying a report in a Jupyter notebook and you see this message, you may need to re-execute the cell or to trust the notebook (button on the top right or "File > Trust notebook").

Now that you defined the predictive model, let’s make a similar analysis than earlier. First, let’s make a sanity check that plot forecast of our model on a subset of the training data to make a sanity check.

# Write your code here.
#
#
#
#
#
#
#
#
#
#
#

altair.Chart(
    pl.concat(
        [
            targets.skb.preview(),
            predictions_ridge.rename(
                {target_column_name: predicted_target_column_name}
            ).skb.preview(),
        ],
        how="horizontal",
    ).tail(24 * 7)
).transform_fold(
    [target_column_name, predicted_target_column_name],
).mark_line(
    tooltip=True
).encode(
    x="prediction_time:T", y="value:Q", color="key:N"
).interactive()

Now, let’s evaluate the performance of the model using cross-validation. Use the time-based cross-validation splitter ts_cv_5 defined earlier. Make sure to compute the R2 score and the mean absolute percentage error. Return the training scores as well as the fitted pipeline such that we can make additional analysis.

Does this model perform better or worse than the previous model? Is it underfitting or overfitting?

# Write your code here.
#
#
#
#
#
#
#
#
#
#
#

cv_results_ridge = predictions_ridge.skb.cross_validate(
    cv=ts_cv_5,
    scoring={
        "r2": get_scorer("r2"),
        "mape": make_scorer(mean_absolute_percentage_error),
    },
    return_train_score=True,
    return_pipeline=True,
    verbose=1,
    n_jobs=-1,
)

[Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.

/home/runner/work/forecasting/forecasting/.pixi/envs/doc/lib/python3.12/site-packages/sklearn/feature_selection/_univariate_selection.py:111: RuntimeWarning: divide by zero encountered in divide
  f = msb / msw

/home/runner/work/forecasting/forecasting/.pixi/envs/doc/lib/python3.12/site-packages/sklearn/feature_selection/_univariate_selection.py:111: RuntimeWarning: divide by zero encountered in divide
  f = msb / msw

/home/runner/work/forecasting/forecasting/.pixi/envs/doc/lib/python3.12/site-packages/sklearn/feature_selection/_univariate_selection.py:111: RuntimeWarning: divide by zero encountered in divide
  f = msb / msw
/home/runner/work/forecasting/forecasting/.pixi/envs/doc/lib/python3.12/site-packages/sklearn/feature_selection/_univariate_selection.py:111: RuntimeWarning: divide by zero encountered in divide
  f = msb / msw

/home/runner/work/forecasting/forecasting/.pixi/envs/doc/lib/python3.12/site-packages/joblib/externals/loky/process_executor.py:782: UserWarning:

A worker stopped while some jobs were given to the executor. This can be caused by a too short worker timeout or by a memory leak.

/home/runner/work/forecasting/forecasting/.pixi/envs/doc/lib/python3.12/site-packages/sklearn/feature_selection/_univariate_selection.py:111: RuntimeWarning: divide by zero encountered in divide
  f = msb / msw

[Parallel(n_jobs=-1)]: Done   5 out of   5 | elapsed:   25.7s finished

Compute all cross-validated predictions to plot the Lorenz curve and the reliability diagram for this pipeline.

To do so, you can use the function collect_cv_predictions to collect the predictions and then call the plot_lorenz_curve and plot_reliability_diagram functions to plot the results.

# Write your code here.
#
#
#
#
#
#
#
#
#
#
#

cv_predictions_ridge = collect_cv_predictions(
    cv_results_ridge["pipeline"], ts_cv_5, predictions_ridge, prediction_time
)

plot_lorenz_curve(cv_predictions_ridge).interactive()

plot_reliability_diagram(cv_predictions_ridge).interactive().properties(
    title="Reliability diagram from cross-validation predictions"
)

Now, let’s perform a randomized search on the hyper-parameters of the model. The code to perform the search is shown below. Since it will be pretty computationally expensive, we are reloading the results of the parallel coordinates plot.

# randomized_search_ridge = predictions_ridge.skb.get_randomized_search(
#     cv=ts_cv_2,
#     scoring="r2",
#     n_iter=100,
#     fitted=True,
#     verbose=1,
#     n_jobs=-1,
# )

# fig = randomized_search_ridge.plot_results().update_layout(margin=dict(l=200))
# write_json(fig, "parallel_coordinates_ridge.json")

fig = read_json("parallel_coordinates_ridge.json")
fig.update_layout(margin=dict(l=200))

We observe that the default values of the hyper-parameters are in the optimal region explored by the randomized search. This is a good sign that the model is well-specified and that the hyper-parameters are not too sensitive to small changes of those values.

We could further assess the stability of those optimal hyper-parameters by running a nested cross-validation, where we would perform a randomized search for each fold of the outer cross-validation loop as below but this is computationally expensive.

# nested_cv_results_ridge = skrub.cross_validate(
#     environment=predictions_ridge.skb.get_data(),
#     pipeline=randomized_search_ridge,
#     cv=ts_cv_5,
#     scoring={
#         "r2": get_scorer("r2"),
#         "mape": make_scorer(mean_absolute_percentage_error),
#     },
#     n_jobs=-1,
#     return_pipeline=True,
# ).round(3)

# nested_cv_results_ridge.round(3)

Predicting multiple horizons with a multi-output model

Usually, it is really common to predict values for multiple horizons at once. The most naive approach is to train as many models as there are horizons. To achieve this, scikit-learn provides a meta-estimator called MultiOutputRegressor that can be used to train a single model that predicts multiple horizons at once.

In short, we only need to provide multiple targets where each column corresponds to an horizon and this meta-estimator will train an independent model for each column. However, we could expect that the quality of the forecast might degrade as the horizon increases.

Let’s train a gradient boosting regressor for each horizon.

from sklearn.multioutput import MultiOutputRegressor

multioutput_predictions = features_with_dropped_cols.skb.apply(
    MultiOutputRegressor(
        estimator=HistGradientBoostingRegressor(random_state=0), n_jobs=-1
    ),
    y=targets.skb.drop(cols=["prediction_time", "load_mw"]).skb.mark_as_y(),
).skb.set_name("multioutput_hgbr")

Now, let’s just rename the columns for the predictions to make it easier to plot the horizon forecast.

target_column_names = [target_column_name_pattern.format(horizon=h) for h in horizons]
predicted_target_column_names = [
    f"predicted_{target_column_name}" for target_column_name in target_column_names
]
named_predictions = multioutput_predictions.rename(
    {k: v for k, v in zip(target_column_names, predicted_target_column_names)}
)

Let’s plot the horizon forecast on a training data to check the validity of the output.

plot_at_time = datetime.datetime(2021, 4, 19, 0, 0, tzinfo=datetime.timezone.utc)
historical_timedelta = datetime.timedelta(hours=24 * 5)
plot_horizon_forecast(
    targets,
    named_predictions,
    plot_at_time,
    historical_timedelta,
    target_column_name_pattern,
).skb.preview()

On this curve, the red line corresponds to the observed values past to the the date for which we would like to forecast. The orange line corresponds to the observed values for the next 24 hours and the blue line corresponds to the predicted values for the next 24 hours.

Since we are using a strong model and very few training data to check the validity we observe that our model perfectly fits the training data.

So, we are now ready to assess the performance of this multi-output model and we need to cross-validate it. Since we do not want to aggregate the metrics for the different horizons, we need to create a scikit-learn scorer in which we set multioutput="raw_values" to get the scores for each horizon.

Passing this scorer to the cross_validate function returns all horizons scores.

from sklearn.metrics import r2_score


def multioutput_scorer(regressor, X, y, score_func, score_name):
    y_pred = regressor.predict(X)
    return {
        f"{score_name}_horizon_{h}h": score
        for h, score in enumerate(
            score_func(y, y_pred, multioutput="raw_values"), start=1
        )
    }


def scoring(regressor, X, y):
    return {
        **multioutput_scorer(regressor, X, y, mean_absolute_percentage_error, "mape"),
        **multioutput_scorer(regressor, X, y, r2_score, "r2"),
    }


multioutput_cv_results = multioutput_predictions.skb.cross_validate(
    cv=ts_cv_5,
    scoring=scoring,
    return_train_score=True,
    verbose=1,
    n_jobs=-1,
)

[Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.

/home/runner/work/forecasting/forecasting/.pixi/envs/doc/lib/python3.12/site-packages/joblib/externals/loky/process_executor.py:782: UserWarning:

A worker stopped while some jobs were given to the executor. This can be caused by a too short worker timeout or by a memory leak.

[Parallel(n_jobs=-1)]: Done   5 out of   5 | elapsed:  1.9min finished

One thing that we observe is that training such multi-output model is expensive. It is expected since each horizon involves a different model and thus a training.

multioutput_cv_results.round(3)

	fit_time	score_time	test_mape_horizon_1h	train_mape_horizon_1h	test_mape_horizon_2h	train_mape_horizon_2h	test_mape_horizon_3h	train_mape_horizon_3h	test_mape_horizon_4h	train_mape_horizon_4h	...	test_r2_horizon_20h	train_r2_horizon_20h	test_r2_horizon_21h	train_r2_horizon_21h	test_r2_horizon_22h	train_r2_horizon_22h	test_r2_horizon_23h	train_r2_horizon_23h	test_r2_horizon_24h	train_r2_horizon_24h
0	76.164	1.826	0.013	0.007	0.020	0.011	0.025	0.012	0.027	0.013	...	0.949	0.993	0.958	0.994	0.959	0.994	0.962	0.995	0.963	0.995
1	83.526	1.785	0.016	0.008	0.025	0.012	0.029	0.014	0.030	0.014	...	0.970	0.992	0.974	0.993	0.975	0.994	0.977	0.994	0.977	0.994
2	84.346	1.520	0.016	0.009	0.023	0.013	0.025	0.015	0.027	0.015	...	0.960	0.991	0.970	0.993	0.971	0.993	0.973	0.993	0.973	0.993
3	83.641	1.825	0.012	0.010	0.018	0.014	0.021	0.015	0.023	0.016	...	0.973	0.989	0.977	0.992	0.980	0.993	0.979	0.993	0.979	0.993
4	20.555	0.457	0.013	0.010	0.019	0.014	0.023	0.016	0.026	0.016	...	0.969	0.989	0.974	0.992	0.976	0.993	0.978	0.993	0.978	0.993

5 rows × 98 columns

Instead of reading the results in the table, we can plot the scores depending on the type of data and the metric.

import itertools
from IPython.display import display

for metric_name, dataset_type in itertools.product(["mape", "r2"], ["train", "test"]):
    columns = multioutput_cv_results.columns[
        multioutput_cv_results.columns.str.startswith(f"{dataset_type}_{metric_name}")
    ]
    data_to_plot = multioutput_cv_results[columns]
    data_to_plot.columns = [
        col.replace(f"{dataset_type}_", "")
        .replace(f"{metric_name}_", "")
        .replace("_", " ")
        for col in columns
    ]

    data_long = data_to_plot.melt(var_name="horizon", value_name="score")
    chart = (
        altair.Chart(
            data_long,
            title=f"{dataset_type.title()} {metric_name.upper()} Scores by Horizon",
        )
        .mark_boxplot(extent="min-max")
        .encode(
            x=altair.X(
                "horizon:N",
                title="Horizon",
                sort=altair.Sort(
                    [f"horizon {h}h" for h in range(1, data_to_plot.shape[1])]
                ),
            ),
            y=altair.Y("score:Q", title=f"{metric_name.upper()} Score"),
            color=altair.Color("horizon:N", legend=None),
        )
    )

    display(chart)

An interesting and unexpected observation is that the MAPE error on the test data is first increases and then decreases once past the horizon 18h. We would not necessarily expect this behaviour.

Native multi-output handling using RandomForestRegressor

In the previous section, we showed how to wrap a HistGradientBoostingRegressor in a MultiOutputRegressor to predict multiple horizons. With such a strategy, it means that we trained independent HistGradientBoostingRegressor, one for each horizon.

RandomForestRegressor natively supports multi-output regression: instead of independently training a model per horizon, it will train a joint model that predicts all horizons at once.

Repeat the previous analysis using a RandomForestRegressor. Fix the parameter min_samples_leaf to 5.

Once you created the model, plot the horizon forecast for a given date and time. In addition, compute the cross-validated predictions and plot the R2 and MAPE scores for each horizon.

Does this model perform better or worse than the previous model?

from sklearn.ensemble import RandomForestRegressor

# Write your code here.
#
#
#
#
#
#
#
#
#
#
#

multioutput_predictions_rf = features_with_dropped_cols.skb.apply(
    RandomForestRegressor(min_samples_leaf=5, random_state=0, n_jobs=-1),
    y=targets.skb.drop(cols=["prediction_time", "load_mw"]).skb.mark_as_y(),
).skb.set_name("multioutput_rf")

named_predictions_rf = multioutput_predictions_rf.rename(
    {k: v for k, v in zip(target_column_names, predicted_target_column_names)}
)

plot_at_time = datetime.datetime(2021, 4, 24, 0, 0, tzinfo=datetime.timezone.utc)
historical_timedelta = datetime.timedelta(hours=24 * 5)
plot_horizon_forecast(
    targets,
    named_predictions_rf,
    plot_at_time,
    historical_timedelta,
    target_column_name_pattern,
).skb.preview()

plot_at_time = datetime.datetime(2021, 4, 25, 0, 0, tzinfo=datetime.timezone.utc)
plot_horizon_forecast(
    targets,
    named_predictions_rf,
    plot_at_time,
    historical_timedelta,
    target_column_name_pattern,
).skb.preview()

multioutput_cv_results_rf = multioutput_predictions_rf.skb.cross_validate(
    cv=ts_cv_5,
    scoring=scoring,
    return_train_score=True,
    verbose=1,
    n_jobs=-1,
)

[Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.

[Parallel(n_jobs=-1)]: Done   5 out of   5 | elapsed:  3.3min finished

multioutput_cv_results_rf.round(3)

	fit_time	score_time	test_mape_horizon_1h	train_mape_horizon_1h	test_mape_horizon_2h	train_mape_horizon_2h	test_mape_horizon_3h	train_mape_horizon_3h	test_mape_horizon_4h	train_mape_horizon_4h	...	test_r2_horizon_20h	train_r2_horizon_20h	test_r2_horizon_21h	train_r2_horizon_21h	test_r2_horizon_22h	train_r2_horizon_22h	test_r2_horizon_23h	train_r2_horizon_23h	test_r2_horizon_24h	train_r2_horizon_24h
0	127.541	0.473	0.024	0.013	0.027	0.014	0.029	0.014	0.031	0.015	...	0.918	0.986	0.918	0.986	0.916	0.985	0.913	0.984	0.908	0.982
1	149.611	0.255	0.029	0.014	0.034	0.015	0.036	0.015	0.037	0.016	...	0.934	0.986	0.937	0.986	0.938	0.985	0.938	0.984	0.936	0.982
2	158.369	0.285	0.023	0.014	0.027	0.015	0.029	0.016	0.030	0.016	...	0.929	0.986	0.933	0.986	0.934	0.985	0.934	0.983	0.931	0.982
3	160.443	0.246	0.019	0.014	0.022	0.016	0.023	0.016	0.025	0.017	...	0.931	0.984	0.933	0.984	0.933	0.983	0.932	0.982	0.930	0.980
4	65.541	0.085	0.025	0.014	0.028	0.016	0.030	0.016	0.032	0.017	...	0.923	0.984	0.925	0.985	0.926	0.984	0.926	0.982	0.923	0.980

5 rows × 98 columns

import itertools
from IPython.display import display

for metric_name, dataset_type in itertools.product(["mape", "r2"], ["train", "test"]):
    columns = multioutput_cv_results_rf.columns[
        multioutput_cv_results.columns.str.startswith(f"{dataset_type}_{metric_name}")
    ]
    data_to_plot = multioutput_cv_results_rf[columns]
    data_to_plot.columns = [
        col.replace(f"{dataset_type}_", "")
        .replace(f"{metric_name}_", "")
        .replace("_", " ")
        for col in columns
    ]

    data_long = data_to_plot.melt(var_name="horizon", value_name="score")
    chart = (
        altair.Chart(
            data_long,
            title=f"{dataset_type.title()} {metric_name.upper()} Scores by Horizon",
        )
        .mark_boxplot(extent="min-max")
        .encode(
            x=altair.X(
                "horizon:N",
                title="Horizon",
                sort=altair.Sort(
                    [f"horizon {h}h" for h in range(1, data_to_plot.shape[1])]
                ),
            ),
            y=altair.Y("score:Q", title=f"{metric_name.upper()} Score"),
            color=altair.Color("horizon:N", legend=None),
        )
    )

    display(chart)

We observe that the performance of the RandomForestRegressor is not better in terms of scores or computational cost. The trend of the scores along the horizon is also different from the HistGradientBoostingRegressor: the scores worsen as the horizon increases.

Uncertainty quantification using quantile regression

In this section, we show how one can use a gradient boosting but modify the loss function to predict different quantiles and thus obtain an uncertainty quantification of the predictions.

from sklearn.metrics import d2_pinball_score

scoring = {
    "r2": get_scorer("r2"),
    "mape": make_scorer(mean_absolute_percentage_error),
    "d2_pinball_05": make_scorer(d2_pinball_score, alpha=0.05),
    "d2_pinball_50": make_scorer(d2_pinball_score, alpha=0.50),
    "d2_pinball_95": make_scorer(d2_pinball_score, alpha=0.95),
}

common_params = dict(
    loss="quantile", learning_rate=0.1, max_leaf_nodes=100, random_state=0
)
predictions_hgbr_05 = features_with_dropped_cols.skb.apply(
    HistGradientBoostingRegressor(**common_params, quantile=0.05),
    y=target,
)
predictions_hgbr_50 = features_with_dropped_cols.skb.apply(
    HistGradientBoostingRegressor(**common_params, quantile=0.5),
    y=target,
)
predictions_hgbr_95 = features_with_dropped_cols.skb.apply(
    HistGradientBoostingRegressor(**common_params, quantile=0.95),
    y=target,
)

cv_results_hgbr_05 = predictions_hgbr_05.skb.cross_validate(
    cv=ts_cv_5,
    scoring=scoring,
    return_pipeline=True,
    verbose=1,
    n_jobs=-1,
)
cv_results_hgbr_50 = predictions_hgbr_50.skb.cross_validate(
    cv=ts_cv_5,
    scoring=scoring,
    return_pipeline=True,
    verbose=1,
    n_jobs=-1,
)
cv_results_hgbr_95 = predictions_hgbr_95.skb.cross_validate(
    cv=ts_cv_5,
    scoring=scoring,
    return_pipeline=True,
    verbose=1,
    n_jobs=-1,
)

[Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.

[Parallel(n_jobs=-1)]: Done   5 out of   5 | elapsed:   10.4s finished

[Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.

[Parallel(n_jobs=-1)]: Done   5 out of   5 | elapsed:   12.8s finished

[Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.

[Parallel(n_jobs=-1)]: Done   5 out of   5 | elapsed:   10.1s finished

cv_results_hgbr_05[
    [col for col in cv_results_hgbr_05.columns if col.startswith("test_")]
].mean(axis=0).round(3)

test_r2               0.853
test_mape             0.051
test_d2_pinball_05    0.698
test_d2_pinball_50    0.644
test_d2_pinball_95   -1.250
dtype: float64

cv_results_hgbr_50[
    [col for col in cv_results_hgbr_50.columns if col.startswith("test_")]
].mean(axis=0).round(3)

test_r2               0.970
test_mape             0.024
test_d2_pinball_05    0.180
test_d2_pinball_50    0.841
test_d2_pinball_95    0.494
dtype: float64

cv_results_hgbr_95[
    [col for col in cv_results_hgbr_95.columns if col.startswith("test_")]
].mean(axis=0).round(3)

test_r2               0.848
test_mape             0.064
test_d2_pinball_05   -2.392
test_d2_pinball_50    0.611
test_d2_pinball_95    0.775
dtype: float64

results = pl.concat(
    [
        targets.skb.select(cols=["prediction_time", target_column_name]).skb.preview(),
        predictions_hgbr_05.rename({target_column_name: "quantile_05"}).skb.preview(),
        predictions_hgbr_50.rename({target_column_name: "median"}).skb.preview(),
        predictions_hgbr_95.rename({target_column_name: "quantile_95"}).skb.preview(),
    ],
    how="horizontal",
).tail(24 * 7)

median_chart = (
    altair.Chart(results)
    .transform_fold([target_column_name, "median"])
    .mark_line(tooltip=True)
    .encode(x="prediction_time:T", y="value:Q", color="key:N")
)

quantile_band_chart = (
    altair.Chart(results)
    .mark_area(opacity=0.4, tooltip=True)
    .encode(
        x="prediction_time:T",
        y="quantile_05:Q",
        y2="quantile_95:Q",
        color=altair.value("lightgreen"),
    )
)

combined_chart = quantile_band_chart + median_chart
combined_chart.interactive()

cv_predictions_hgbr_05 = collect_cv_predictions(
    cv_results_hgbr_05["pipeline"], ts_cv_5, predictions_hgbr_05, prediction_time
)
cv_predictions_hgbr_50 = collect_cv_predictions(
    cv_results_hgbr_50["pipeline"], ts_cv_5, predictions_hgbr_50, prediction_time
)
cv_predictions_hgbr_95 = collect_cv_predictions(
    cv_results_hgbr_95["pipeline"], ts_cv_5, predictions_hgbr_95, prediction_time
)

plot_residuals_vs_predicted(cv_predictions_hgbr_05).interactive().properties(
    title=(
        "Residuals vs Predicted Values from cross-validation predictions"
        " for quantile 0.05"
    )
)

plot_residuals_vs_predicted(cv_predictions_hgbr_50).interactive().properties(
    title=(
        "Residuals vs Predicted Values from cross-validation predictions" " for median"
    )
)

plot_residuals_vs_predicted(cv_predictions_hgbr_95).interactive().properties(
    title=(
        "Residuals vs Predicted Values from cross-validation predictions"
        " for quantile 0.95"
    )
)

cv_predictions_hgbr_05_concat = pl.concat(cv_predictions_hgbr_05, how="vertical")
cv_predictions_hgbr_50_concat = pl.concat(cv_predictions_hgbr_50, how="vertical")
cv_predictions_hgbr_95_concat = pl.concat(cv_predictions_hgbr_95, how="vertical")

import matplotlib.pyplot as plt
from sklearn.metrics import PredictionErrorDisplay


for kind in ["actual_vs_predicted", "residual_vs_predicted"]:
    fig, axs = plt.subplots(1, 3, figsize=(15, 5), sharey=True)

    PredictionErrorDisplay.from_predictions(
        y_true=cv_predictions_hgbr_05_concat["load_mw"].to_numpy(),
        y_pred=cv_predictions_hgbr_05_concat["predicted_load_mw"].to_numpy(),
        kind=kind,
        ax=axs[0],
    )
    axs[0].set_title("0.05 quantile regression")

    PredictionErrorDisplay.from_predictions(
        y_true=cv_predictions_hgbr_50_concat["load_mw"].to_numpy(),
        y_pred=cv_predictions_hgbr_50_concat["predicted_load_mw"].to_numpy(),
        kind=kind,
        ax=axs[1],
    )
    axs[1].set_title("Median regression")

    PredictionErrorDisplay.from_predictions(
        y_true=cv_predictions_hgbr_95_concat["load_mw"].to_numpy(),
        y_pred=cv_predictions_hgbr_95_concat["predicted_load_mw"].to_numpy(),
        kind=kind,
        ax=axs[2],
    )
    axs[2].set_title("0.95 quantile regression")

    fig.suptitle(f"{kind} for GBRT minimzing different quantile losses")

def coverage(y_true, y_quantile_low, y_quantile_high):
    y_true = np.asarray(y_true)
    y_quantile_low = np.asarray(y_quantile_low)
    y_quantile_high = np.asarray(y_quantile_high)
    return float(
        np.logical_and(y_true >= y_quantile_low, y_true <= y_quantile_high)
        .mean()
        .round(4)
    )


def mean_width(y_true, y_quantile_low, y_quantile_high):
    y_true = np.asarray(y_true)
    y_quantile_low = np.asarray(y_quantile_low)
    y_quantile_high = np.asarray(y_quantile_high)
    return float(np.abs(y_quantile_high - y_quantile_low).mean().round(1))

coverage(
    cv_predictions_hgbr_50_concat["load_mw"].to_numpy(),
    cv_predictions_hgbr_05_concat["predicted_load_mw"].to_numpy(),
    cv_predictions_hgbr_95_concat["predicted_load_mw"].to_numpy(),
)

mean_width(
    cv_predictions_hgbr_50_concat["load_mw"].to_numpy(),
    cv_predictions_hgbr_05_concat["predicted_load_mw"].to_numpy(),
    cv_predictions_hgbr_95_concat["predicted_load_mw"].to_numpy(),
)

# Compute binned coverage scores
binned_coverage_results = binned_coverage(
    [df["load_mw"].to_numpy() for df in cv_predictions_hgbr_50],
    [df["predicted_load_mw"].to_numpy() for df in cv_predictions_hgbr_05],
    [df["predicted_load_mw"].to_numpy() for df in cv_predictions_hgbr_95],
    n_bins=10,
)

binned_coverage_results

	bin_left	bin_right	bin_center	fold_idx	coverage	mean_width	n_samples
0	28744.0	36884.0	32814.0	0	0.4205	4235.5	459
1	36886.0	40158.0	38522.0	0	0.6493	4291.2	499
2	40160.0	42982.0	41571.0	0	0.7117	4501.7	496
3	42983.0	45219.0	44101.0	0	0.7400	4281.1	473
4	45220.0	47332.0	46276.0	0	0.7401	4071.5	454
5	47334.0	49718.0	48526.0	0	0.7044	3868.6	477
6	49720.0	53126.0	51423.0	0	0.7418	4679.3	426
7	53129.0	57570.0	55349.5	0	0.8162	6144.5	321
8	57572.0	63173.0	60372.5	0	0.8687	7106.8	259
9	63176.0	86573.0	74874.5	0	0.8929	8807.9	168
10	28744.0	36884.0	32814.0	1	0.4637	3917.9	496
11	36886.0	40158.0	38522.0	1	0.7960	4557.4	402
12	40160.0	42982.0	41571.0	1	0.8550	4842.4	407
13	42983.0	45219.0	44101.0	1	0.8783	4546.3	378
14	45220.0	47332.0	46276.0	1	0.8939	4620.4	377
15	47334.0	49718.0	48526.0	1	0.8648	5101.1	392
16	49720.0	53126.0	51423.0	1	0.8450	5774.8	400
17	53129.0	57570.0	55349.5	1	0.8657	6392.7	402
18	57572.0	63173.0	60372.5	1	0.8005	6515.6	386
19	63176.0	86573.0	74874.5	1	0.6505	7550.9	392
20	28744.0	36884.0	32814.0	2	0.6887	4985.7	318
21	36886.0	40158.0	38522.0	2	0.7657	5111.9	367
22	40160.0	42982.0	41571.0	2	0.7731	5186.4	357
23	42983.0	45219.0	44101.0	2	0.8050	4952.7	400
24	45220.0	47332.0	46276.0	2	0.8814	4906.7	413
25	47334.0	49718.0	48526.0	2	0.8951	4984.1	410
26	49720.0	53126.0	51423.0	2	0.8716	5976.0	475
27	53129.0	57570.0	55349.5	2	0.8577	6568.6	534
28	57572.0	63173.0	60372.5	2	0.8386	6386.9	446
29	63176.0	86573.0	74874.5	2	0.6731	7098.5	312
30	28744.0	36884.0	32814.0	3	0.7563	4199.4	513
31	36886.0	40158.0	38522.0	3	0.8534	3682.2	498
32	40160.0	42982.0	41571.0	3	0.8358	3972.7	481
33	42983.0	45219.0	44101.0	3	0.8337	3750.7	457
34	45220.0	47332.0	46276.0	3	0.8767	3718.1	511
35	47334.0	49718.0	48526.0	3	0.7968	4168.4	497
36	49720.0	53126.0	51423.0	3	0.7019	4535.3	369
37	53129.0	57570.0	55349.5	3	0.8386	5236.3	254
38	57572.0	63173.0	60372.5	3	0.7041	5945.4	267
39	63176.0	86573.0	74874.5	3	0.3568	7160.5	185
40	28744.0	36884.0	32814.0	4	0.6217	4230.8	230
41	36886.0	40158.0	38522.0	4	0.7570	4435.0	251
42	40160.0	42982.0	41571.0	4	0.8723	4569.2	274
43	42983.0	45219.0	44101.0	4	0.8350	4213.3	309
44	45220.0	47332.0	46276.0	4	0.8385	4167.3	260
45	47334.0	49718.0	48526.0	4	0.7137	4547.9	241
46	49720.0	53126.0	51423.0	4	0.8324	5691.7	346
47	53129.0	57570.0	55349.5	4	0.8591	6481.9	504
48	57572.0	63173.0	60372.5	4	0.7857	6359.0	658
49	63176.0	86573.0	74874.5	4	0.5495	6589.4	959

coverage_by_bin = binned_coverage_results.copy()
coverage_by_bin["bin_label"] = coverage_by_bin.apply(
    lambda row: f"[{row.bin_left:.0f}, {row.bin_right:.0f}]", axis=1
)

ax = coverage_by_bin.boxplot(
    column="coverage", by="bin_label", figsize=(12, 6), vert=False, whis=1000
)
ax.axvline(x=0.9, color="red", linestyle="--", label="Target coverage (0.9)")
ax.set_xlabel("Load bins (MW)")
ax.set_ylabel("Coverage")
ax.set_title("Coverage Distribution by Load Bins")
ax.legend()
plt.suptitle("")  # Remove automatic suptitle from boxplot
plt.xticks(rotation=45)
plt.tight_layout()

Reliability diagrams and Lorenz curves for quantile regression

plot_reliability_diagram(
    cv_predictions_hgbr_50, kind="quantile", quantile_level=0.50
).interactive().properties(
    title="Reliability diagram for quantile 0.50 from cross-validation predictions"
)

plot_reliability_diagram(
    cv_predictions_hgbr_05, kind="quantile", quantile_level=0.05
).interactive().properties(
    title="Reliability diagram for quantile 0.05 from cross-validation predictions"
)

plot_reliability_diagram(
    cv_predictions_hgbr_95, kind="quantile", quantile_level=0.95
).interactive().properties(
    title="Reliability diagram for quantile 0.95 from cross-validation predictions"
)

plot_lorenz_curve(cv_predictions_hgbr_50).interactive().properties(
    title="Lorenz curve for quantile 0.50 from cross-validation predictions"
)

plot_lorenz_curve(cv_predictions_hgbr_05).interactive().properties(
    title="Lorenz curve for quantile 0.05 from cross-validation predictions"
)

plot_lorenz_curve(cv_predictions_hgbr_95).interactive().properties(
    title="Lorenz curve for quantile 0.95 from cross-validation predictions"
)

Quantile regression as classification

In the following, we turn a quantile regression problem for all possible quantile levels into a multiclass classification problem by discretizing the target variable into bins and interpolating the cumulative sum of the bin membership probability to estimate the CDF of the distribution of the continuous target variable conditioned on the features.

Ideally, the classifier should be efficient when trained on a large number of classes (induced by the number of bins). Therefore we use a Random Forest classifier as the default base estimator.

There are several advantages to this approach:

• a single model is trained and can jointly estimate quantiles for all quantile levels (assuming a well tuned number of bins);

• the quantile levels can be chosen at prediction time, which allows for a flexible quantile regression model;

• in practice, the resulting predictions are often reasonably well calibrated as we will see in the reliability diagrams below.

One possible drawback is that current implementations of gradient boosting models tend to be very slow to train with a large number of classes. Random Forests are much more efficient in this case, but they do not always provide the best predictive performance. It could be the case that combining this approach with tabular neural networks can lead to competitive results.

However, the current scikit-learn API is not expressive enough to to handle the output shape of the quantile prediction function. We therefore cannot make it fit into a skrub pipeline.

from scipy.interpolate import interp1d
from sklearn.base import BaseEstimator, RegressorMixin, clone
from sklearn.utils.validation import check_is_fitted
from sklearn.ensemble import RandomForestClassifier
from sklearn.preprocessing import KBinsDiscretizer
from sklearn.utils.validation import check_consistent_length
from sklearn.utils import check_random_state
import numpy as np


class BinnedQuantileRegressor(BaseEstimator, RegressorMixin):
    def __init__(
        self,
        estimator=None,
        n_bins=100,
        quantile=0.5,
        random_state=None,
    ):
        self.n_bins = n_bins
        self.estimator = estimator
        self.quantile = quantile
        self.random_state = random_state

    def fit(self, X, y):
        # Lightweight input validation: most of the input validation will be
        # handled by the sub estimators.
        random_state = check_random_state(self.random_state)
        check_consistent_length(X, y)
        self.target_binner_ = KBinsDiscretizer(
            n_bins=self.n_bins,
            strategy="quantile",
            subsample=200_000,
            encode="ordinal",
            quantile_method="averaged_inverted_cdf",
            random_state=random_state,
        )

        y_binned = (
            self.target_binner_.fit_transform(np.asarray(y).reshape(-1, 1))
            .ravel()
            .astype(np.int32)
        )

        # Fit the multiclass classifier to predict the binned targets from the
        # training set.
        if self.estimator is None:
            estimator = RandomForestClassifier(random_state=random_state)
        else:
            estimator = clone(self.estimator)
        self.estimator_ = estimator.fit(X, y_binned)
        return self

    def predict_quantiles(self, X, quantiles=(0.05, 0.5, 0.95)):
        check_is_fitted(self, "estimator_")
        edges = self.target_binner_.bin_edges_[0]
        n_bins = edges.shape[0] - 1
        expected_shape = (X.shape[0], n_bins)
        y_proba_raw = self.estimator_.predict_proba(X)

        # Some might stay empty on the training set. Typically, classifiers do
        # not learn to predict an explicit 0 probability for unobserved classes
        # so we have to post process their output:
        if y_proba_raw.shape != expected_shape:
            y_proba = np.zeros(shape=expected_shape)
            y_proba[:, self.estimator_.classes_] = y_proba_raw
        else:
            y_proba = y_proba_raw

        # Build the mapper for inverse CDF mapping, from cumulated
        # probabilities to continuous prediction.
        y_cdf = np.zeros(shape=(X.shape[0], edges.shape[0]))
        y_cdf[:, 1:] = np.cumsum(y_proba, axis=1)
        return np.asarray([interp1d(y_cdf_i, edges)(quantiles) for y_cdf_i in y_cdf])

    def predict(self, X):
        return self.predict_quantiles(X, quantiles=(self.quantile,)).ravel()

quantiles = (0.05, 0.5, 0.95)
bqr = BinnedQuantileRegressor(
    RandomForestClassifier(
        n_estimators=300,
        min_samples_leaf=5,
        max_features=0.2,
        n_jobs=-1,
        random_state=0,
    ),
    n_bins=30,
)
bqr

BinnedQuantileRegressor(estimator=RandomForestClassifier(max_features=0.2,
                                                         min_samples_leaf=5,
                                                         n_estimators=300,
                                                         n_jobs=-1,
                                                         random_state=0),
                        n_bins=30)

In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.

from sklearn.model_selection import cross_validate

X, y = features_with_dropped_cols.skb.eval(), target.skb.eval()

cv_results_bqr = cross_validate(
    bqr,
    X,
    y,
    cv=ts_cv_5,
    scoring={
        "d2_pinball_50": make_scorer(d2_pinball_score, alpha=0.5),
    },
    return_estimator=True,
    return_indices=True,
    verbose=1,
    n_jobs=-1,
)

[Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.

[Parallel(n_jobs=-1)]: Done   5 out of   5 | elapsed:  2.1min finished

cv_predictions_bqr_all = [
    cv_predictions_bqr_05 := [],
    cv_predictions_bqr_50 := [],
    cv_predictions_bqr_95 := [],
]
for fold_ix, (qreg, test_idx) in enumerate(
    zip(cv_results_bqr["estimator"], cv_results_bqr["indices"]["test"])
):
    print(f"CV iteration #{fold_ix}")
    print(f"Test set size: {test_idx.shape[0]} rows")
    print(
        f"Test time range: {prediction_time.skb.eval()[test_idx][0, 0]} to "
        f"{prediction_time.skb.eval()[test_idx][-1, 0]} "
    )
    y_pred_all_quantiles = qreg.predict_quantiles(X[test_idx], quantiles=quantiles)

    coverage_score = coverage(
        y[test_idx],
        y_pred_all_quantiles[:, 0],
        y_pred_all_quantiles[:, 2],
    )
    print(f"Coverage: {coverage_score:.3f}")

    mean_width_score = mean_width(
        y[test_idx],
        y_pred_all_quantiles[:, 0],
        y_pred_all_quantiles[:, 2],
    )
    print(f"Mean prediction interval width: " f"{mean_width_score:.1f} MW")

    for q_idx, (quantile, predictions) in enumerate(
        zip(quantiles, cv_predictions_bqr_all)
    ):
        observed = y[test_idx]
        predicted = y_pred_all_quantiles[:, q_idx]
        predictions.append(
            pl.DataFrame(
                {
                    "prediction_time": prediction_time.skb.eval()[test_idx],
                    "load_mw": observed,
                    "predicted_load_mw": predicted,
                }
            )
        )
        print(f"d2_pinball score: {d2_pinball_score(observed, predicted):.3f}")
    print()

CV iteration #0
Test set size: 4032 rows
Test time range: 2023-02-11 00:00:00+00:00 to 2023-07-28 23:00:00+00:00 
Coverage: 0.965
Mean prediction interval width: 10778.0 MW
d2_pinball score: 0.346
d2_pinball score: 0.735
d2_pinball score: -0.025

CV iteration #1
Test set size: 4032 rows
Test time range: 2023-07-29 00:00:00+00:00 to 2024-01-12 23:00:00+00:00 

Coverage: 0.992
Mean prediction interval width: 11109.8 MW
d2_pinball score: 0.360
d2_pinball score: 0.813
d2_pinball score: 0.275

CV iteration #2
Test set size: 4032 rows
Test time range: 2024-01-13 00:00:00+00:00 to 2024-06-28 23:00:00+00:00 
Coverage: 0.988
Mean prediction interval width: 11762.3 MW
d2_pinball score: 0.292
d2_pinball score: 0.789
d2_pinball score: 0.108

CV iteration #3
Test set size: 4032 rows

Test time range: 2024-06-29 00:00:00+00:00 to 2024-12-13 23:00:00+00:00 
Coverage: 0.989
Mean prediction interval width: 9291.9 MW
d2_pinball score: 0.248
d2_pinball score: 0.806
d2_pinball score: 0.316

CV iteration #4
Test set size: 4032 rows
Test time range: 2024-12-14 00:00:00+00:00 to 2025-05-30 23:00:00+00:00 

Coverage: 0.978
Mean prediction interval width: 13170.0 MW
d2_pinball score: 0.349
d2_pinball score: 0.790
d2_pinball score: 0.249

# Let's assess the calibration of the quantile regression model:

plot_reliability_diagram(
    cv_predictions_bqr_50, kind="quantile", quantile_level=0.50
).interactive().properties(
    title="Reliability diagram for quantile 0.50 from cross-validation predictions"
)

plot_reliability_diagram(
    cv_predictions_bqr_05, kind="quantile", quantile_level=0.05
).interactive().properties(
    title="Reliability diagram for quantile 0.05 from cross-validation predictions"
)

plot_reliability_diagram(
    cv_predictions_bqr_95, kind="quantile", quantile_level=0.95
).interactive().properties(
    title="Reliability diagram for quantile 0.95 from cross-validation predictions"
)

We can complement this assessment with the Lorenz curves, which only assess the ranking power of the predictions, irrespective of their absolute values.

plot_lorenz_curve(cv_predictions_bqr_50).interactive().properties(
    title="Lorenz curve for quantile 0.50 from cross-validation predictions"
)

plot_lorenz_curve(cv_predictions_bqr_05).interactive().properties(
    title="Lorenz curve for quantile 0.05 from cross-validation predictions"
)

plot_lorenz_curve(cv_predictions_bqr_95).interactive().properties(
    title="Lorenz curve for quantile 0.95 from cross-validation predictions"
)